Lecture notes on accumulation theories

Lecture notes on accumulation theories

The mather (or father) of all growth models: Harrod- Domar’s model and the instability problem

• Dual nature of investment: it generates capacity, it generates

aggregate demand. But more while more investment to day solve the current problem of capacity utilization, it creates new capacity, increasing the problem tomorrow.

• As Domar (1947, pp. 49-50) put it:

• “The economy finds itself in a serious dilemma: if sufficient

investment is not forthcoming today, unemployment will be here today. But if enough is invested today, still more will be needed tomorrow. … So that far as unemployment is concerned,

investment is at the same time a cure for the disease and the cause of even greater ills in the future. “

Domar’s model

Conclusions on Domar’s model

Harrod’s model:

• S1 = sY₁ (1) (multiplier)

• I₁ = v_n (Y_1E - Y₀) (2) (accelerator)

• S₁ = I₁ (3) (equilibrium in the commodity

market: there is dynamic equilibrium if I₁ generates a growth of AD and capacity such as to generate an identical S₁)

• v_n = K/Y = K/Y is the desired (or required) capital coefficient, s marginal propensity to save

• "The axiomatic basis of the theory which I propose to develop consists of three propositions, namely: (a) that the level of a

community income is the most important determinant of its supply of saving; (b) that the rate of increase of its income is an important

determinant of its demand for saving; and (c) that demand is equal to supply. It thus consists in a marriage of the 'acceleration principle' and the 'multiplier' theory" (Harrod, 1939, p.43).

The warranted rate

• By simple substitution we get:

• v_n (Y_1E - Y₀) = sY₁

• or g_w = (Y_1E - Y₀) /Y₁ = s/v_n

• gw = s/v_n is the rate of growth at which capacity savings (savings forthcoming from existing capacity utilised at its normal or ‘expected level) is equal to investment.

• Note that g_w = gE, that is the economy will grow in equilibrium (I = capacity saving) if the entrepreneurs expect a rate of growth (gE) equal to gw.

The warranted rate and the instability question

• gw = s/v_n, is the so-called warranted growth rate . "The warranted rate of growth is taken to be that rate of growth which, if it occurs, will

leave all parties satisfied that they have produced neither more nor less than the right amount" (1939, p.45).

• gE = s/v_n , that is if the entrepreneurs expect a growth rate g_E = g_w = s/v_n, then the economy grows with a goods market equilibrium (g_A = g_E)

• If the entrepreneurs knew g_w, they would decide an investment level that would precisely generate g_w. But they do not know it.

• Instability:

• If gE > g_W  g_A > g_E and next time they will expect an even larger G_E

• If g_E < g_W  g_A < g_E and next time they will expect an even lower G_E

•

A simple example

• Suppose v = 2, s = 0.2 so that gw = 0.1 (or 10%).

• Suppose current output is 90, if ge = g_w = 0.1, firms would invest 20 (= 2 x 10) and output at t+1 will indeed be 100 (we take the growth rate as a proportion of final output), and S = I.

• If ge > g_w, and they expect, say, 101, firms will invest 22 and output will be 110, so investors will feel that they were too pessimistic (ga >

ge > g_w  0.2 > 0.11 > 0.1) and revise their expectations upward.

• So they will revise ge upward and ga will diverge even more from gw.

• Similarly if they anticipate 99, they will invest 18 and output at t+1 will be only 90. They will therefore feel that they were too aptimistic and revise their expectations downward, so the decline will

continue.

Problems with Harrod

• Harrod’s model raised three questions:

• g_w is not a full employment path (in general g_w ≠ n): this is a problem only for neoclassical economists

• Capitalists do not know g_w

• gw is unstable, that is the economy does not gravitate around it (not surprisingly, unless you believe in Say’s Law, there is no reason why in Harrod I = capacity savings).

• Next step: Instability seen through the lenses of the degree of capacity utilisation

• Normal degree of capacity utilisation : the average degree of capacity utilisation desired by the entrepreneurs

Degree of capacity utilisation and the capital coefficient

u_n= Y_n/Y_f where Y_f is the maximum physical output from a given capacity K. In general Y_n < Y_f and u_n < u_max = 1. The main reason (Steindl) is that firms want to keep a margin of non utilised capacity to meet

sudden peaks of demand; if peaks persist, they will increase capacity.

When gA > gw, it means that s/va> s/vn,, that is v_a = K/Y_a < v_n = K/Y_n  Y_a > Y_n

or, in terms of degree of capacity utilisation u, u_a = Y_a/Y_f > u_n = Y_n/Y_f  Y_a > Y_n

Read in the opposite direction: whenever gA > gw, Ya > Yn ua > un, the actual degree of capacity utilisation u_a is higher than normal (so

entrepreneurs invest even more to restore u_n , but in this way they aggravate the disequilibrium).

The opposite would of course happen when g_A < g_w (u_a < u_n and investment would keep falling to absorb the less-than-normal u).

Recent survey:

Attilio Trezzini and Daria Pignalosa, The Normal Degree of Capacity Utilization: The History of a Controversial Concept, CSWP 49 (April 2021)

Anticipation

• To anticipate what will follow, when we shall consider the heterodox growth models, one way to accommodate Harrod’s instability is to say: if g_A > g_w and both v_a < v_n and u_a > u_n , then we take v_a and u_a as the “new normal” capital coefficient and degree of capacity utilisation. But this sounds a post hoc ergo propter hoc kind of

reasoning adopted by the (decreasingly) influential ‘neo-Kaleckian’

growth models.

Two roads to get out from Harrod’s troubles

• Harrod influential in the 1950s (the market economy is unstable, then the State must plan the economy: France’s planning, in Italy Piano Vanoni, India’s planning). This is normative, but Harrod is unsatisfactory as a positive explanation, markets are unstable, but not explosive.

• Note a “traditional” feature of Harrod’s model: s positively influences gw and growth is endogenous (in the sense that it depends on

preferences about present and future consumption). Perfect for a neoclassical economist, were not for the absence of full

employment. Solow’s neoclassical growth model (1956) recovers full employment, but losses, somewhat paradoxically, the influence of s on g. Endogenous growth theory (that actually begun in the early 1960s) is just on having both.

• Two early ways out form Harrod’s problems: Solow’s neoclassical growth model and the Cambridge Equation (later followed by other heterodox schools)

Two roads to get out from Harrod’s troubles II

• Main characteristic of the neoclassical approach is the dependence of investment from savings, more specifically from full-capacity

savings, savings that spring from a productive capacity that is both fully (or normally) employed, and also fully employs all the labour supply.

• Main characteristic of (most of ) the alternative approaches is what Garegnani named (after Kaldor) the “Keynesian Hypothesis”

(KH) that investment are independent from savings both in the short and in the long run.*

• This is the striking difference between neoclassical economists of all persuasions (including neo-Keynesians a là Blanchard, Stiglitz,

Krugman, De Long etc): aggregate demand determines output both in the short and in the long period, or more precisely, it is capacity that adjusts to aggregate demand both in the short and in the long period. We shall begin from neoclassical growth story.

• * However, there are a number of “heterodox models” that are

“keynesian” in the short-run and “Classical” (in the sense of Say’s

After Harrod Harrod

Solow Cambridge equation

EGT Neo-kaleckian

and/versus Sraffians

Before we consider some heterodox models, let us remind some critical aspects of neoclassical growth theory

• Solow’s fundamental equation:

•

k n sy

k   (  )

 

The Solowian warranted growth rate

• At k*, sy = ( + n)k, and k=0.

• Note that in equilibrium g_w = sy/k= (+ n), that is s/v* = ( + n), where v* is the equilibrium value of v.

• (Hidden behind any equlibrium growth equation there is Harrods’s g_w)

• Neglecting , gw = n, a stable, full employment path

• In Solow the actual growth rate tends to/gravitates towards the warranted rate

• This depends on «productive factors’» substitutability

Gravitation towards the equilibrium (real and financial mechanisms) On the left of k*: sy > ( + n)k, i falls, k rises.

On the right of k*: sy < ( + n)k, i rises, k falls.

• “In its basic form the neo-classical model depends on the assumption that it is always possible and consistent with equilibrium that investment should be undertaken of an amount equal to full-employment savings. The mechanism that ensures this is as a rule not specified. Most neo-classical writers have, however, had in mind some financial mechanism. In the ideal neo-classical world one may think of there being a certain level of the rate of interest (r) that will lead entrepreneurs, weighing interest cost against expected profits, to carry out investment equal to full-employment savings.” Hahn & Metthews EJ dec. 1964, my italics.

• On the left of k*, since sy > ( + n)k, then sy/k > ( + n), and given that k/y = va, g^a = s/va > ( + n), where va is the actual value of v.

• va has to rise to realise the neoclassical warranted equilibrium path s/ v* = ( + n).

• And indeed (from the left) the increase of k implies a rise of v = y/k=f(k)/k  next slide

Gravitation towards the equilibrium (real and financial mechanisms) On the left of k*: sy > ( + n)k, i falls, k rises.

On the right of k*: sy < ( + n)k, i rises, k falls.

• “In its basic form the neo-classical

model depends on the assumption that it is always possible and consistent with equilibrium that investment should be undertaken of an amount equal to full-employment savings. The

mechanism that ensures this is as a rule not specified. Most neo-classical writers have, however, had in mind some financial mechanism. In the ideal neo-classical world one may think of there being a certain level of the rate of interest (r) that will lead

entrepreneurs, weighing interest cost against expected profits, to carry out investment equal to full-employment savings.” Hahn & Metthews EJ dec.

1964, my italics.

The neoclassical adjustment through the lenses of the Harrodian equation

• On the left of k*, since sy >

( + n)k, then sy/k > ( + n), and given that k/y = va, g^a = s/va > ( + n), where va is the actual value of v.

• va has to rise to realise the neoclassical warranted

equilibrium path s/ v* = ( + n).

• And indeed (from the left) the increase of k implies a rise of v

= y/k=f(k)/k  next slide

Solowian stability

• Why (on the left of the stationary state k*) when k rises,then v also rises (so that ga = s/v falls)?

• va = k/y: when k rises,also y rises, but given the falling marginal productivity of capital, it rises less than proportionally than k. So v rises and va  v* (A visual inspection is enough to see that, for a given k, y is progressively smaller).

• we see that the neocl. subst. mech.s entail the gravitation towards a full employment path: the fall in the interest rate induces the

entrepreneurs to adopt more capital intensive techniques

Solowian stability

• Note that this result depends upon what Hahn & Metthews pointed out, a decreasing demand function for investment (capital)

negatively elastic to the interest rate. The capital theory controversy has shown that this function can have any shape. So Solow’s model is not robust, it’s deadly wrong.

• Seen from a different perspective: there is no rigorous foundation to the proposition that a falling interst rate induce firms to adopt more capital intensive techniques. So no increase of k when we are on the left of k*. Rigorously, k can rise, then fall, then rise again. No convergence, no stability, no full employment growth.

Comparative statics: a change in s entails higher y and k but not a higher g_w: a disappointing result (it came as a surprise)

Just a word about Endogenous Growth Theory

• Many empirical studies show that there is a positive correlation between I/Y (the investment share) and g (the growth rate).

• Through neoclassical lenses (according to which I = S) this is a correlation between s = S/Y and g.

• But in Solow a rise of s only affect y not g!.

• From the very early sixties this is the start of EGT: a desperate attempt to establish a connection between s and G.

• Not surprisingly, most of EGT is a return to Harrod (g is

“endogenous” in Harrod, endogenous in the specific marginalist sense that it depends on s, on preferences between present and future consumption) (Solow dixit in 1992 in this room!)

• If curious, please see my papers on EGT (see my web page)